The Hartree equation for infinitely many particles. I. Well-posedness theory

Abstract

We show local and global well-posedness results for the Hartree equation $$i{\partial }_t\gamma =[-\Delta +w\ast {\rho }_{\gamma },\gamma ],$$ where $\gamma$ is a bounded self-adjoint operator on $L^2({\mathbb{R}}^d)$, ${\rho }_{\gamma }(x)=\gamma (x,x)$ and $w$ is a smooth short-range interaction potential. The initial datum $\gamma (0)$ is assumed to be a perturbation of a translation-invariant state ${\gamma }_f=f(-\Delta )$ which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state $\gamma (t)$, counted relatively to the stationary state ${\gamma }_f$. We indeed use a general notion of relative entropy, which allows to treat a wide class of stationary states $f(-\Delta )$. Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.

to appear in Comm. Math. Phys

References in corpus (4)

• Mean-field Evolution of Fermionic Systems
• The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D
• A Nonlinear Model for Relativistic Electrons at Positive Temperature
• Energy Cost to Make a Hole in the Fermi Sea

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